Fundamentals of Digital Image Processing - Control of Color Imaging Systems: Analysis and Design - page 29

f a ( x , y ) f* ( x , y )

s ( x , y )

FIGURE 2.8 Sampling process.

1 1

s ( x, y ) ¼ d( x n D x, y m D y ) ( 2 : 30 )

n ¼1 m ¼1

1
D x

1

The sampling frequencies in the x- and y-directions are f s x ¼ and f s y ¼

D y ,

respectively.
The sampling process is basically the multiplication of f a ( x, y ) and s ( x, y ) ,as

shown in Figure 2.8.

1 1

f * ( x, y ) ¼ f a ( n D x , m D y )d( x n D x , y m D y ) ( 2 : 31 )

n ¼1 m ¼1

The spectrum of the sampled signal is obtained by taking the Fourier transform
of f * ( x, y ) . That is,

1 1

( x, y ) e j (v x x þv y y ) d x d y F* (v x , v y ) ¼ f *

1 1

1 1

1 1

d( x n D x , y m D y ) e j (v x x þv y y ) d x d y ¼ f a ( n D x , m D y )

n ¼1 m ¼1

1 1

1 1

f ( n, m ) e j (v x n D x þv y m D y ) ¼ ( 2 : 32 )

n ¼1 m ¼1

where f ( n, m ) ¼ f a ( n D x , m D y ) . Since f a ( x, y ) can be obtained by taking the inverse

Fourier transform of F (v x , v y ) , then we have

1 1

1

v y ) e j (v x x þv y y ) dv x dv y j x ¼ n D x ,y ¼ m D y f ( n, m ) ¼ f a ( x, y )j x ¼ n D x ,y ¼ m D y ¼ F (v x ,

4 p

2

1 1

1 1

1

v y ) e j (v x n D x þv y m D y ) dv x dv y ¼ F (v x , ( 2 : 33 )

4 p

2

1 1